Using The Refinement Equation For The Construction of Pre-Wavelets VI: Shift Invariant Subspaces

This paper follows the format of our tutorial on multivariate wavelet decomposition, [6]. We demonstrate here that the methods employed by Jia and Micchelli [2] can be extended to subspaces of L 2(ℝ s ) generated by a finite number of compactly supported functions. Specifically, we let Φ: ℝ s → ℂN be a vector of N functions Φ = (φ1,…,φ N ). AS in our previous article in these proceedings we will be concerned with wavelet decomposition in L 2(ℝs) built upon Φ and the basic operations of shift and scale. All the results in [2] pertain to the case N = 1. We now proceed to extend them to general N.